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In-Depth Information
CHAPTER FOUR
The Statistical Significance of
F
and Effect Strength
We continue here with our discussion of the elements of the summary
table provided in Table 3.2. We have discussed the sources of variance, the
sum of squares, the degrees of freedom, and the mean squares in Chap-
ter 3; we are now ready to deal with the
F
ratio, the probability of the
F
value occurring by chance based on its sampling distribution (so that
we can determine whether or not it is statistically significant), and the
eta squared value (
2
) representing the strength of the effect of the
independent variable. We begin with the
F
ratio.
η
4.1 THE
F
RATIO
4.1.1 THE
F
RATIO AND THE NULL HYPOTHESIS
As Snedecor (1946) stated so well, ANOVA tests the null hypothesis, which
states that the observed means for the various conditions in the study rep-
resent “ . . . random samples from the same normal population” (p. 219).
Expressed in more conversational language, if the null hypothesis is true,
then the means of the conditions in a study are not, statistically, signif-
icantly different. An extremely convenient way to test the null hypoth-
esis is to compute the ratio of the between-groups variance (the mean
square value associated with the between-subjects source of variance) to
the within-groups variance (the mean square value associated with the
within-subjects source of variance). This may be written as follows:
between-groups
MS
within-groups
MS
MS
A
MS
S
/
A
.
F
=
=
(4.1)
In our example study (see Table 3.2), the
F
ratio is 18.75. To evaluate this
ratio, it is necessary to know how the ratio is distributed based on chance.
And that brings us back to R. A. Fisher.
4.1.2 THE
F
RATIO AND R. A. FISHER
Howtheratioofmeansquarebetweengroups(
MS
A
)tomeansquare
within groups (
MS
S
/
A
) is distributed based on chance was worked out
and presented to an international mathematics conference by R. A. Fisher
in 1924 (Kirk, 1995). Because this ratio plays such a pivotal role in ANOVA,
it was bound to take on a name early in its history. The name that stuck
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